By tensoring the map \(R\mathcal {H}om(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet })_{0}\rightarrow \tilde{\pi }_{\widetilde{\mathbf {M}}^s}^*\mathbb {L}_{\widetilde{\mathbf {M}}^s/S}[1]\)  with \(\tilde{\pi }_{\mathcal {X}}^*\omega _{\mathcal {X}/S}\) , and using Serre duality in [[1]}, Theorem 2.22 and Corollary 2.10], we have the following map
\(\widetilde{\Phi }: \widetilde{\mathbf {E}}^\bullet :=R\tilde{\pi }_{\widetilde{\mathbf {M}}^s*}(R\mathcal {H}om(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet })_{0}\otimes \tilde{\pi }_{\mathcal {X}}^*\omega _{\mathcal {X}/S})[2]\rightarrow \mathbb {L}_{\widetilde{\mathbf {M}}^s/S}.\) 
